# Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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### Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...

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### Computing a cone in a $\otimes$-triangulated category

I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$:
$$
x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0.
$$
Consider the ...

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### Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'

It appears to me (though I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as ...

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### Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?

Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...

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### Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...

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### Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory:
Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...

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### Are there universal homological functors?

There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...

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### Abelianization derivator

About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....

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### Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...

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### Filtered triangulated category examples

I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category ...

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### Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...

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### Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...

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### When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...

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### Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...

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### Highest weight category and weight structures

In various branches of representation theory, there is a notion of highest weight category.
On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced by ...

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### Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...

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### Why do we need the negative sign in TR2 (the “turning triangle” axiom) in the definition of triangulated categories?

In the TR2 of the definition of triangulated categories, we add a negative sign to an arrow when we turn the triangles. What is the significance/motivation of that negative sign ($-u[1]$ as in the ...

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### Vanishing natural transformation exact triangle

This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...

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### Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...

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### If the homotopy category is well-generated, must the $\infty$-category be presentable?

Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...

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### A question on Voevodsky´s categories

I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.-...

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### (Middling) good morphisms of triangles

Neeman in his article "Some new axioms for triangulated categories" calls a morphism of distinguished triangles
$$\require{AMScd}
\begin{CD}
X @>>> Y @>>> Z @>>> X [1] \\
@...

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151 views

### Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group

There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...

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### Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...

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### Triangulated categories generated by a collection of submodules

In the book "Cohomology Rings of Finite Groups", by Carlson- Townsley - Elizondo there is the following corollary
The category $\mathsf{stmod}_{\mathbb{k}G}$ is generated as a triangulated category ...

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### Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory

Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...

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### Extending a natural transformation using a distinguished triangle

$\require{AMScd}$
Let $\mathcal{T}$ be a triangulated category,
and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated).
Let $F, G: \mathcal{T} \to \mathcal{T}$ be two ...

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### Techniques for Showing Triviality of K_1 of a Higher Category

Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are ...

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### Grothendieck group of limit of categories

I am in the following situation. I have a stable presentable $\infty$-category $\cal{C}$, and a sequence of full stable subcategories $\dots\subset\cal{C}_{-2}\subset\cal{C}_{-1}\subset\cal{C}_0\...

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### Thick subcategories

I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...

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### the cohomology objects in the t-structure and long exact sequence

It is known that, given an abelian category $\mathcal B$, the derived category $\mathrm D (\mathcal B)$ is a triangulated category. In particular, given a distinguished triangle $E\to F\to G \to E[1]$ ...

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### Subcategories of the Verdier quotient?

Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.
Is there a bijective correspondence between ...

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### Does stabilization preserve the property of being a full subcategory?

Suppose $F:\mathcal{C}\hookrightarrow \mathcal{D}$ is a fully faithful symmetric monoidal functor of symmetric monoidal $\infty$-categories. Suppose $x\in\mathcal{C}$. Is it true that $\mathcal{C}[x^{-...

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### Are mapping cones in the bounded homotopy category of chain complexes isomorphic?

Let $A$ be an additive category. Suppose we have distinguished triangles
$$X \rightarrow Y \rightarrow Z \rightarrow X[1]$$
and
$$X \rightarrow Y \rightarrow Z' \rightarrow X[1]$$
in the bounded ...

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### How to get that one module is tilting iff the other one is?

Let $A$ be an algebra over a field k. $D$ is the standard duality functor. A module $_AM$ is called a generator if $add(A) \subseteq add(M)$, a cogenerator if $add(D(A)) \subseteq add(M)$. $M$ is n-...

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### Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence

Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...

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### Is the (bounded) derived category of a smooth projective scheme locally finite?

In the paper "Locally finite triangulated categories" Xiao and Zhu define locally finite triangulated categories. These are triangulated categories $\mathcal{A}$ such that
$$\sum_{X \in \mathrm{ind}\...

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### Do we have the contravariant Hom exact sequence in a pretriangulated category?

Let $\mathcal{C}$ be a pretriangulated dg-category over $k$. By definition, we call $X\to Y\to Z\to X[1]$ an exact triangle in $\mathcal{C}$ if for any $W$, $C(W,Z)$ is homotopic to cone$(\mathcal{C}(...

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### Brown Representability for Stable Homotopy Categories of Symmetric Spectra

Proposition 5.5 in $\mathbf{A}^1$-homotopy theory establishes the Brown's representability for the stable homotopy category $\mathcal{SH}_T(S)$, over a Noetherian scheme $S$, for a space $T$ of finite ...

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### A characterization of distinguished triangles in triangulated categories.

Let $\mathscr{D}$ be a triangulated category. Let $$X \longrightarrow Y \longrightarrow Z \longrightarrow X[1]$$
be a triangle (not necessarily distinguished). We call it special if for each $E \in \...

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### relation between extensions and $\mathrm{Hom}$ for an abelian subcategory of a triangulated category

Let $T$ be a triangulated category and $A$ and abelian full subcategory of $T$. Consider the Yoneda extension groups $Ext_{A}^n$. For any two objects $X$ and $Y$ of $A$ here is a map
$$
Ext^n_A(X, Y) ...

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### $\mathrm{Hom}_D(D^{\geq 1},D^{\leq 0})=0?$

Let $D$ be a triangulated category with $t$-structure given by strictly full subcategories $D^{\leq0},D^{\geq 1}$. By definition we have $\mathrm{Hom}_D(D^{\leq 0},D^{\geq 1})=0$.
Can we decuce from ...

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### Are endofunctors of triangulated categories triangulated?

Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{A}:=\operatorname{End}(\mathcal{T})$ be the category of triangulated endofunctors of $\mathcal{T}$.
Is $\mathcal{A}$ triangulated?
My ...

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### On the definition of triangulated categories

Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category ...

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### Objects of which Grothendieck abelian categories have elements?

The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those ...

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### On various relations between “additional axioms” for AB4 and Grothendieck abelian categories

Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list ...

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### Which triangulated categories are subcategories of compact objects “somewhere”?

Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of ...

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### On countable homotopy colimits in (the derived categories of) AB3 abelian categories

If $h_i:A_i\to A_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A_i$ in $D^b(A)$. This is a two-term ...

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### Triangulated category with finitely many indecomposables and all objects have period 4

Let a triangulated category always mean k-linear over a field k and locally finite and Krull-Schmidt in the following.
Is there a list of triangulated categories with finitely many indecomposables ...

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### Examples of $1$-Calabi-Yau triangulated categories

Can you give me examples of $1$-Calabi-Yau triangulated categories $D$ different from the bounded derived category of coherent sheaves on an elliptic curve? I would like moreover the numerical ...